Hamiltonian Equations Research Articles

Higher-order contact mechanics

We present a complete theory of higher-order autonomous contact mechanics, which allows us to describe higher-order dynamical systems with dissipation. The essential tools for the theory are the extended higher-order tangent bundles, TkQ×R, whose geometric structures are previously introduced in order to state the Lagrangian and Hamiltonian formalisms for these kinds of systems, including their variational formulation. The variational principle, the contact forms, and the geometric dynamical equations are obtained by using those structures and generalizing the standard formulation of contact Lagrangian and Hamiltonian systems. As an alternative approach, we develop a unified description that encompasses the Lagrangian and Hamiltonian equations as well as their relationship through the Legendre map; all of them are obtained from the contact dynamical equations and the constraint algorithm that is implemented because, in this formalism, the dynamical systems are always singular. Some interesting examples are finally analysed using these geometric formulations.

Numeric‐analytical solution of an LC circuit with nonlinear capacitor based on the symplectic conservative perturbation method

AbstractThe LC circuit with a nonlinear capacitor is analyzed using the symplectic conservative perturbation method based on the symplectic matrix. By treating the electric charge and the flux linkage as dual variables, the LC circuit system can be transformed into a Hamiltonian system. Then we solve the nonlinear Hamiltonian matrix equation by the perturbation method. The Hamiltonian matrix can be cast into linear and nonlinear parts. Based on the exact solution of the linear part, the nonlinear part is solved perturbatively by a canonical transformation. Since the coefficient matrix of the obtained equation is still a Hamiltonian matrix, symplectic conservation is guaranteed. The validity of the method is demonstrated, and the stability, precision, and the efficiency of the method are discussed by comparing it with the fourth‐order Runge–Kutta method (RK4), the ordinary perturbation method, the small‐parameter perturbation method based on displacement (SPPD), and the harmonic balance (HB) method. Numerical results show that the proposed method has better stability, precision, and efficiency compared with the other methods and thus has great advantages in long‐term simulations.

Conformal Hamiltonian Mechanical Equations on Contact 9- Manifolds

In this study, we concluded the Hamiltonian equations on, being a model. Finally, introduce, some geometrical and physical results on the related mechanic systems have been discussed.

Hamiltonian Monte Carlo method for estimating variance components.

A Hamiltonian Monte Carlo algorithm is a Markov chain Monte Carlo method, and the method has a potential to improve estimating parameters effectively. Hamiltonian Monte Carlo is based on Hamiltonian dynamics, and it follows Hamilton's equations, which are expressed as two differential equations. In the sampling process of Hamiltonian Monte Carlo, a numerical integration method called leapfrog integration is used to approximately solve Hamilton's equations, and the integration is required to set the number of discrete time steps and the integration stepsize. These two parameters require some amount of tuning and calibration for effective sampling. In this study, we applied the Hamiltonian Monte Carlo method to animal breeding data and identified the optimal tunings of leapfrog integration for normal and inverse chi-square distributions. Then, using real pig data, we revealed the properties of the Hamiltonian Monte Carlo method with the optimal tuning by applying models including variance explained by pedigree information or genomic information. Compared with the Gibbs sampling method, the Hamiltonian Monte Carlo method had superior performance in both models. We have provided the source codes of this method written in the Fortran language at https://github.com/A-ARAKAWA/HMC.

The principle of virtual work and Hamilton's principle on Galilean manifolds

To describe time-dependent finite-dimensional mechanical systems, their generalized space-time is modeled as a Galilean manifold. On this basis, we present a geometric mechanical theory that unifies Lagrangian and Hamiltonian mechanics. Moreover, a general definition of force is given, such that the theory is capable of treating nonpotential forces acting on a mechanical system. Within this theory, we elaborate the interconnections between classical equations known from analytical mechanics such as the principle of virtual work, Lagrange's equations of the second kind, Hamilton's equations, Lagrange's central equation, Hamel's generalized central equation as well as Hamilton's principle.

Taylor Series and Getting the General Solutions for the Equations of Motion Using Poisson Bracket Relations

In mathematics and classical mechanics, the Poisson bracket is an important binary operation in Hamiltonian mechanics, playing a central role in Hamilton's equations of motion, which govern the time evolution of a Hamiltonian dynamical system. The Poisson bracket also distinguishes a certain class of coordinate transformations, called canonical transformations, which map canonical coordinate systems into canonical coordinate systems. In this work we study some examples from the classical mechanics of particles and apply mathematical method for building the equation of motion. In the present paper Poisson Brackets and their properties are presented, by using Poisson brackets and their properties we calculate some brackets. We use the Poisson bracket with Hamiltonians to express the time dependence of a function u (t), the main idea Taylor series is taken as the required solution for equation of motion using the properties of the Poisson Brackets, We have examined examples from the classical mechanics to illustrate the idea such as motion with a constant acceleration, simple harmonic oscillator, freely falling particle. The solutions are compatible with what is known in classical mechanics. The work is fundamental and sheds new light onto classical mechanics. Poisson brackets are a powerful and sophisticated tool in the Hamiltonian formalism of Classical Mechanics. They also happen to provide a direct link between classical and quantum mechanics.

Emergency Autonomous Steering Control Research Based on Receding Horizon H∞ Control and Minimax Criteria

Aiming at the problems of model uncertainty, external interference of road conditions, and inherent nonlinearity of lateral movement of autonomous driving vehicles, this paper proposes an effective control technology for an automatic steering system, namely, the receding horizon $H_{\infty }$ control, which has good path tracking performance and can overcome external interference. First, a two-degree-of-freedom vehicle model and a road model are established. Then, the Hamilton equation is established using Pontriagin’s minimum principle, and the optimal control solution and the most serious disturbance are solved based on the minimax principle. Finally, in order to evaluate the receding horizon $H_{\infty }$ autonomous steering control method, lane change simulation and hardware in the loop simulation are developed to describe the effect of receding horizon $H_{\mathrm {\infty }}$ . The simulation results show that the model predictive control (MPC) and linear matrix inequality (LMI) controller will exhibit oscillation and instability under the condition of external interference. To improve the stability, the $H_{\infty }$ controller reduces part of the performance, while the rolling horizon $H_{\mathrm {\infty }}$ strategy can ensure the robustness of automatic steering and has a certain anti-interference performance.

$ M- $truncated optical soliton and their characteristics to a nonlinear equation governing the certain instabilities of modulated wave trains

<abstract> This study investigates the nonlinear Hamiltonian amplitude equation by using two analytical techniques, namely; the extended sinh-Gordon equation expansion method, and the extended rational sine-cosine/sinh-cosh methods. Some important wave solutions are successfully constructed, such as dark, bright, combined dark-bright, singular solitons, periodic and singular periodic wave solutions. The physical features of the acquired solutions are plotted to depict the clear dynamical behaviour of the reported results. All the acquired solutions have satisfied the original equation. </abstract>

ВИБРОЗАЩИТНЫЕ СИСТЕМЫ ТРАНСПОРТНЫХ СРЕДСТВ: АКТИВНОЕ УПРАВЛЕНИЕ И ОПТИМИЗАЦИЯ

The article presents the results of studies of the dynamic properties of a vibration-protective system with active control, which is directly identified with the compensatory effect and is charac-terized as direct control, as well as a vibration-protective system with indirect control, which forms a compensatory effect mediated through the implementation of an intermittent process of changes in the parameters of the inertial link. To find the optimal positional control function of direct (ac-tive) control, as well as indirect (singular) control of the inertial link, information technologies for solving boundary value problems were used in relation to the canonical Hamilton equations for the vibration protection systems under study. The parameters are established at which vibration protection systems of this type eliminate resonance phenomena and provide comparable indicators of vibration safety for protected objects under kinematic disturbance.

Identification of the infrasound signals emitted by explosive eruption of Mt. Shinmoedake by three-dimensional ray tracing.

Mt. Shinmoedake, a part of the Mt. Kirishima cluster of volcanoes in Kyushu, Japan, erupted on 10 March 2018. Our infrasound sensor network located at a distance of more than 200 km from the source detected signals emitted by an explosive eruption of Mt. Shinmoedake. The arrival time of the signals is divided into three time intervals. To reveal how the observed infrasound signals propagated from the source to the sensors, we carry out three-dimensional ray tracing on the basis of the Hamilton equations including the vertical profiles of the temperature and wind around the ray path. We present formulas for calculating travel time and distance of infrasound from a source to an observation site and its turning altitude in the atmosphere. We have identified four kinds of signals, namely, the waves propagated in the troposphere undergoing multiple refraction and those refracting from the stratosphere, the mesosphere, and the lower thermosphere. Brief discussion is devoted to some of the unidentified signals.

Analiza kretanja i stabilnosti holonomnog mehaničkog sistema u proizvoljnom polju sila

In order to give an insight into the work of the machine before the production and assembly and to obtain good analysis, this paper presents detailed solutions to the specific problem occured in the field of analytical mechanics. In addition to numerical procedures in the paper, a review of the theoretical foundations was made.Various types of analysis are very common in mechanical engineering, due to the possibility of an approximation of complex machines. For the proposed system, Lagrange's equations of the first kind, covariant and contravariant equations, Hamiltons equations and the generalized coordinates, as well as insight in Coulumb friction force are provided.Also, the conditions of static equilibrium are solved numerically and using intersection of the two curves. Finally, stability of motion for the disturbed and undisturbed system was investigated.

Nonlinear vibration and dynamic response of nano composite conical tube by conveying fluid flow

Modeling of tubes containing fluid flow is widely used in the study of heat exchangers, nuclear reactors, micro and nano tools, etc. This system is structurally simple but very complex in terms of dynamic behavior and vibrations. In this paper, an analytical relationship for nonlinear vibrations of self-excitation of a nanocomposite conical tube containing fluid flow is extracted, one end of which is free and the other side is fixed and is under gravitational force. The base material is assumed to be 1200 series aluminum, which is reinforced with carbon nanotubes. The Hamiltonian equations are obtained, assuming the Euler–Bernoulli beam theory and the use of the Galerkin method, dissected the partial derivative equations into Ordinary Differential Equations (ODE), then solved by MATLAB coding and investigated the effect of various parameters on system behavior. As the fluid velocity increases, the amplitude of the vibration increases and the nonlinear effects of the system increase, so more modes are needed to converge the responses. In a conical tube, the βT coefficient increases with increasing inner diameter along the tube and the system becomes more stable. Increasing the length of the pipe makes the opening conical pipe more stable and the closing conical pipe more unstable. The change in length has no effect on the stability of the cylindrical tube.

Explicit high-order energy-preserving methods for general Hamiltonian partial differential equations

A novel class of explicit high-order energy-preserving methods are proposed for general Hamiltonian partial differential equations with non-canonical structure matrix. When the energy is not quadratic, it is firstly done that the original system is reformulated into an equivalent form with a modified quadratic energy conservation law by the energy quadratization approach. Then the resulting system that satisfies the quadratic energy conservation law is discretized in time by combining explicit high-order Runge–Kutta methods with orthogonal projection techniques. The proposed schemes are shown to share the order of explicit Runge–Kutta method and thus can reach the desired high-order accuracy. Moreover, the methods are energy-preserving and explicit because the projection step can be solved explicitly. Numerical results are addressed to demonstrate the remarkable superiority of the proposed schemes in comparison with other structure-preserving methods.

Energy Levels and Electromagnetic Transition of 90-94mo Nuclei Using Ibm-1

The energy states for the J , b , ɤ bands and electromagnetic transitions B (E2) values for even – even molybdenum 90 – 94 Mo nuclei are calculated in the present work of "the interacting boson model (IBM-1)" . The parameters of the equation of IBM-1 Hamiltonian are determined which yield the best excellent suit the experimental energy states . The positive parity of energy states are obtained by using IBS1. for program for even 90 – 94 Mo isotopes with bosons number 5 , 4 and 5 respectively. The" reduced transition probability B(E2)" of these neuclei are calculated and compared with the experimental data . The ratio of the excitation energies of the 41+ to 21+ states ( R4/2) are also calculated . The calculated and experimental (R4/2) values showed that the 90 – 94 Mo nuclei have the vibrational dynamical symmetry U(5). Good agreement was found from comparison between the calculated energy states and electric quadruple probabilities B(E2) transition of the 90–94Mo isotopes with the experimental data .

Quasi-geometric integration of guiding-center orbits in piecewise linear toroidal fields

A numerical integration method for guiding-center orbits of charged particles in toroidal fusion devices with three-dimensional field geometry is described. Here, high order interpolation of electromagnetic fields in space is replaced by a special linear interpolation, leading to locally linear Hamiltonian equations of motion with piecewise constant coefficients. This approach reduces computational effort and noise sensitivity, while the conservation of total energy, magnetic moment and phase space volume is retained. The underlying formulation treats motion in piecewise linear fields exactly and, thus, preserves the non-canonical symplectic form. The algorithm itself is only quasi-geometric due to a series expansion in the orbit parameter. For practical purposes, an expansion to the fourth order retains geometric properties down to computer accuracy in typical examples. When applied to collisionless guiding-center orbits in an axisymmetric tokamak and a realistic three-dimensional stellarator configuration, the method demonstrates stable long-term orbit dynamics conserving invariants. In Monte Carlo evaluation of transport coefficients, the computational efficiency of quasi-geometric integration is an order of magnitude higher than with a standard fourth order Runge–Kutta integrator.

Analytical mechanics allows novel vistas on mathematical epidemic dynamics modeling

This contribution aims to shed light on mathematical epidemic dynamics modelling from the viewpoint of analytical mechanics. To set the stage, it recasts the basic SIR model of mathematical epidemic dynamics in an analytical mechanics setting. Thereby, it considers two possible re-parameterizations of the basic SIR model. On the one hand, it is proposed to re-scale time, while on the other hand, to transform the coordinates, i.e.\ the independent variables. In both cases, Hamilton's equations in terms of a suited Hamiltonian as well as Hamilton's principle in terms of a suited Lagrangian are considered in minimal and extended phase and state space coordinates, respectively. The corresponding Legendre transformations relating the various options for the Hamiltonians and Lagrangians are detailed. Ultimately, this contribution expands on a multitude of novel vistas on mathematical epidemic dynamics modelling that emerge from the analytical mechanics viewpoint. As result, it is believed that interesting and relevant new research avenues open up when exploiting in depth the analogies between analytical mechanics and mathematical epidemic dynamics modelling.

Analysis of crack parameters under pure Mode II loading by modified exponential matrix method

A new analytical method for calculating the Stress Intensity Factor (SIF) under Mode II loading is presented in the framework of two-dimensional stationary elasticity. Starting from the linear elasticity equations, the crack problem is formulated as a system of Hamiltonian equations, solved based on a new method for determining eigensystems, called Exponential Matrix Method (EMM). The obtained eigensolutions satisfy the adjoint symplectic orthogonality by means of performing an angular stress variation with respect to the radial stress. The linear combination of these eigensolutions then allows a good description of the expected solution. Related to one of the asymptotic expansion coefficients, the SIF under Mode II Loading is calculated using the trapezoidal integration method (TIM). The implemented methodology validation is carried out by comparing the obtained results with those provided by the literature review.

Enhanced photon blockade via driving a trapped Λ -type atom in a hybrid optomechanical system

A scheme is proposed to enhance the photon blockade effect in a hybrid optomechanical system with a $\mathrm{\ensuremath{\Lambda}}$-type atom driven by the microwave. Through analyzing the conventional and unconventional blockade mechanisms, we find that the enhanced photon blockade effect can be attributed to two aspects: (i) The nonresonant coupled $\mathrm{\ensuremath{\Lambda}}$-type atom reconstructs the anharmonic eigenenergy spectrum and (ii) the microwave driving field promotes the destructive quantum interference for two-photon excitation. By means of the joint enhancement effect, the perfect photon blockade, i.e., the second-order correlation function ${g}^{(2)}(0)\ensuremath{\simeq}0$, can be achieved without the strong single-photon optomechanical coupling as reported in the standard optomechanical system. All the analyses and derivations are further verified via simulating numerically the quantum master equation of the initial Hamiltonian, showing good agreement between analytical and numerical results. Moreover, the optimal parameter relation is given to optimize the photon blockade and maximize the occupancy probability of single-photon excitation at the same time. Our scheme provides a feasible method to engineer a high-quality and efficient single-photon source.

First Principles of the Classical Mechanics and the Foundations of Statistical Mechanics on the Example of a Disordered Spin System

We study the classical multicomponent disordered 3D spin system taking into account the temperature of the medium in the framework of the model of nearest neighbors. The latter allows the 3D problem with a cubic lattice to reduce to the 1D Heisenberg spin glass problem with a random environment. Using the Hamilton equations of motion, a recurrent equation is obtained that connects three spins in successive nodes of 1D lattice, taking into account the influence of a random environment. This equation, together with the corresponding conditions of a local minimum energy in nodes, allows to construct node-by-node a stable spin chains and, accordingly, to calculate all parameters of statistical ensemble from the first principles of classical mechanics, without using any additional assumptions, in particular, the main axiom of statistical mechanics – the equiprobability of statistical states. Using the example of 1D Heisenberg spin glass model, the features of the new approach are studied in detail and the statistical mechanics of the system are constructed without using the standard representation of the partition function (PF).

Drivetrain fatigue strength characteristics of model‐predictive control for wind turbines

AbstractWind turbines play a crucial role in the revolution towards renewable resources. They need to be economically competitive to be sustainable. This still requires to lower the cost of energy (COE). To this end, nonlinear model predictive control (NMPC) is used within this paper. As known from literature, NMPC significantly improves the energy extracting performance as well as the mitigation of tower loads of wind turbines. As it is shown in this paper, the drivetrain fatigue strength drops disproportionally in parallel, which either increases the demands on the turbine design or decreases the lifetime of drivetrain components. Without additional boundary conditions to the underlying optimal control problems (OCPs), the application of energy‐maximizing NMPC might so even raise the COE. Only penalizing axial torque oscillations by quadratic terms decreases energy‐extracting performance below the level of classical wind turbine controllers. This makes more sophisticated conditions necessary. In this paper, the increase of the drivetrain damage by NMPC is analyzed, and appropriate boundary conditions are derived, to balance the two contradicting objectives of energy maximization and drivetrain load mitigation. An NMPC approach based on indirect methods is used, to obtain a solution of the OCPs very efficiently. It applies the Hamilton equations and Pontryagin's maximum principle. Its accuracy and efficiency to solve OCPs was presented over the last decades.